Note on q-Extensions of Euler Numbers and Polynomials of Higher Order

In 2007, Ozden et al. constructed generating functions of higher-order twisted (h, q)-extension of Euler polynomials and numbers, by using p-adic, q-deformed fermionic integral on . By applying their generating functions, they derived the complete sums of products of the twisted (h, q)-extension of Euler polynomials and numbers. In this paper, we consider the new q-extension of Euler numbers and polynomials to be different which is treated by Ozden et al. From our q-Euler numbers and polynomials, we derive some interesting identities and we construct q-Euler zeta functions which interpolate the new q-Euler numbers and polynomials at a negative integer. Furthermore, we study Barnes-type q-Euler zeta functions. Finally, we will derive the new formula for "sums of products of q-Euler numbers and polynomials" by using fermionic q-adic, q-integral on .